Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 497
i.e. whether the equilibrium point is stable or unstable. Thus, a ZVS is clearly inappropriate, or
at least potentially dangerous, for the discussion of the classical motion close to equilibrium. As a
matter of fact, this di culty with the ZVS is crucial in our case, even for the pure CP case, without
additional magnetic eld. Indeed, the ZVS becomes
2q +1
S = − x
2
2 q − 1
+
2 y2
! 2 ; (223)
where we use the single parameter q to parametrize S. Atq = 1 (i.e., F = 0, see Eq. (217)), the
equilibrium point turns from a saddle (for q¿1) into a maximum. Consequently, the ZVS correctly
re ects the change of the equilibrium point from unstable (q¿1) to stable (q¡1). However,
for any q ∈ (0; 1), the equilibrium remains a maximum of the ZVS, which completely misses the
change of stability at q =8=9. Thus, the very same maximum may change its stability (which
fundamentally a ects the classical motion in its vicinity) without being noticed by inspection of
the ZVS. The latter evolves very smoothly around q =8=9. Thus, the ZVS contours provide no
information on the nature of the classical motion in the vicinity of the equilibrium point, in disaccord
with [30,62,44,46,54,144,151]. Similarly, the isovalue contours of the ZVS (which are ellipses in the
harmonic approximation) have no relation with the isovalue contours of the ground-state wave packet
localized around the equilibrium point (these contours are also ellipses in the harmonic approximation
where the wave packet is a Gaussian), contrary to what is stated in [30,44]. For example, the aspect
ratio (major axis=minor axis) of the ZVS contour lines is (2q +1)=(1 − q) which varies smoothly
around q =8=9, while the aspect ratio of the isocontours of the ground state wave-packet diverges
when q → 8=9. 25
Nonwithstanding, a ZVS may be used for other purposes [144], e.g., to show the existence of
an ionization threshold for the Hamiltonian (214), when !c ¿! (area coded in black in Fig. 39).
Clearly, due to the parabolic con nement in the x–y plane, ionization is only possible along the z
direction. The threshold is given by [144]
Eion = F 2 =2!(! − !c) ; (224)
which lies above the equilibrium energy Eeq. Thus, for parameters in that region, the electron—
initially placed close to the stable xed point—cannot ionize. One may expect, therefore, that wave
packets built around the equilibrium point for !c ¿! lead to discrete Floquet states. In other
cases, e.g., for pure CP driving, non-dispersive wave packets are rather represented by long-living
resonances (see Section 7.1).
Finally, it has been often argued [30,44,46,54,62,144,151] that the presence of the magnetic eld
is absolutely necessary for the construction of non-dispersive wave packets. The authors consider
non-dispersive wave packets as equivalent to Gaussian-shaped wave functions (using equivalently the
notion of coherent states). Then it is vital that the motion in the vicinity of the xed point is locally
harmonic within a region of size ˝. This leads the authors to conclude that non-dispersive wave
packets may not exist for the pure CP case except in the extreme semiclassical regime. As opposed
to that, the diamagnetic term in Eq. (214) gives a stronger weight to the harmonic term, which is
25
While this argument has been presented here for the simplest case of the harmonic oscillator Hamiltonian (191), it
carries over to the full, non-linear model, Eq. (214).